Coverage for local/lib/python2.7/site-packages/sage/tensor/differential_forms.py : 58%

r""" 

Algebra of differential forms 

Algebra of differential forms defined on a CoordinatePatch (an open subset of 

Euclidian space, see ``CoordinatePatch`` for details). 

AUTHORS: 

- Joris Vankerschaver (2010-05-26) 

.. TODO:: 

- Allow for forms with values in a vector space 

- Incorporate Kahler differentials 

REFERENCES: 

- R. Abraham, J. E. Marsden, and T. S. Ratiu: Manifolds, tensor analysis, 

and applications. Springer-Verlag 1988, texts in Applied Mathematical 

Sciences, volume 75, 2nd edition. 

- :wikipedia:`Differential_form` 

""" 

#***************************************************************************** 

# Copyright (C) 2010 Joris Vankerschaver (joris.vankerschaver@gmail.com) 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from six.moves import range 

from sage.rings.ring import Algebra 

from sage.tensor.coordinate_patch import CoordinatePatch 

from sage.tensor.differential_form_element import DifferentialForm 

from sage.symbolic.ring import SR, var 

class DifferentialForms(Algebra): 

""" 

The algebra of all differential forms on an open subset of Euclidian space 

of arbitrary dimension. 

EXAMPLES: 

To define an algebra of differential forms, first create a coordinate 

patch:: 

sage: p, q = var('p, q') 

sage: U = CoordinatePatch((p, q)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^2 with coordinates p, q 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables p, q 

If no coordinate patch is supplied, a default one (using the variables 

x, y, z) will be used:: 

sage: F = DifferentialForms(); F 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

""" 

Element = DifferentialForm 

def __init__(self, coordinate_patch = None): 

""" 

Construct the algebra of differential forms on a given coordinate patch. 

See ``DifferentialForms`` for details. 

INPUT: 

- ``coordinate_patch`` -- Coordinate patch where the algebra lives. 

If no coordinate patch is given, a default coordinate patch with 

coordinates (x, y, z) is used. 

EXAMPLES:: 

sage: p, q = var('p, q') 

sage: U = CoordinatePatch((p, q)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^2 with coordinates p, q 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables p, q 

""" 

from sage.categories.graded_algebras_with_basis \ 

import GradedAlgebrasWithBasis 

from sage.structure.parent_gens import ParentWithGens 

from sage.misc.superseded import deprecation 

deprecation(24444, 'For the set of differential forms of degree p, ' + 

'use U.diff_form_module(p), where U is the base ' + 

'manifold (type U.diff_form_module? for details).') 

if not coordinate_patch: 

x, y, z = var('x, y, z') 

coordinate_patch = CoordinatePatch((x, y, z)) 

if not isinstance(coordinate_patch, CoordinatePatch): 

raise TypeError("%s not a valid Coordinate Patch" % coordinate_patch) 

self._patch = coordinate_patch 

ParentWithGens.__init__(self, SR, \ 

category = GradedAlgebrasWithBasis(SR)) 

def __eq__(self, other): 

""" 

Return True if self is equal to other. 

EXAMPLES:: 

sage: x, y, z = var('x, y, z') 

sage: U = CoordinatePatch((x, y, z)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^3 with coordinates x, y, z 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: p, q = var('p, q') 

sage: V = CoordinatePatch((p, q)); V 

Open subset of R^2 with coordinates p, q 

sage: G = DifferentialForms(V); G 

Algebra of differential forms in the variables p, q 

sage: H = DifferentialForms(U); H 

Algebra of differential forms in the variables x, y, z 

sage: F == G 

False 

sage: F == H 

True 

""" 

if type(other) is type(self): 

return self._patch == other._patch 

else: 

return False 

def __ne__(self, other): 

""" 

Return True if self is not equal to other. 

EXAMPLES:: 

sage: x, y, z = var('x, y, z') 

sage: U = CoordinatePatch((x, y, z)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^3 with coordinates x, y, z 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: p, q = var('p, q') 

sage: V = CoordinatePatch((p, q)); V 

Open subset of R^2 with coordinates p, q 

sage: G = DifferentialForms(V); G 

Algebra of differential forms in the variables p, q 

sage: F != G 

True 

""" 

return not self == other 

def ngens(self): 

""" 

Return the number of generators of this algebra. 

EXAMPLES:: 

sage: x, y, z = var('x, y, z') 

sage: U = CoordinatePatch((x, y, z)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^3 with coordinates x, y, z 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: F.ngens() 

3 

""" 

return len(self._patch.coordinates()) 

def gen(self, i=0): 

""" 

Return the `i^{th}` generator of ``self``. This is a one-form, 

more precisely the exterior derivative of the i-th coordinate. 

INPUT: 

- ``i`` - integer (optional, default 0) 

EXAMPLES:: 

sage: x, y, z = var('x, y, z') 

sage: U = CoordinatePatch((x, y, z)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^3 with coordinates x, y, z 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: F.gen(0) 

doctest:...: DeprecationWarning: Use U.diff_form(degree) instead, 

where U is the base manifold (type U.diff_form? for details). 

See http://trac.sagemath.org/24444 for details. 

dx 

sage: F.gen(1) 

dy 

sage: F.gen(2) 

dz 

""" 

form = DifferentialForm(self, 0, self._patch.coordinate(i)) 

return form.diff() 

def gens(self): 

""" 

Return a list of the generators of ``self``. 

EXAMPLES:: 

sage: x, y, z = var('x, y, z') 

sage: U = CoordinatePatch((x, y, z)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^3 with coordinates x, y, z 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: F.gens() 

(dx, dy, dz) 

""" 

return tuple(self.gen(n) for n in range(self._patch.dim())) 

def base_space(self): 

""" 

Return the coordinate patch on which this algebra is defined. 

EXAMPLES:: 

sage: x, y, z = var('x, y, z') 

sage: U = CoordinatePatch((x, y, z)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^3 with coordinates x, y, z 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: F.base_space() 

Open subset of R^3 with coordinates x, y, z 

""" 

return self._patch 

def _element_constructor_(self, fun): 

""" 

Coerce a given function (element of the symbolic ring) 

into a differential form of degree zero. 

EXAMPLES:: 

sage: x, y, z = var('x, y, z') 

sage: U = CoordinatePatch((x, y, z)) 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: F(sin(x*y)) # indirect doctest 

doctest:...: DeprecationWarning: Use U.diff_form(degree) instead, 

where U is the base manifold (type U.diff_form? for details). 

See http://trac.sagemath.org/24444 for details. 

sin(x*y) 

""" 

fun = SR(fun) 

if fun not in self: 

raise ValueError("Function not an element of this algebra of differential forms.") 

return DifferentialForm(self, 0, fun) 

def __contains__(self, element): 

""" 

Check if a given element belongs to this algebra of differential forms. 

EXAMPLES:: 

sage: x, y, p, q = var('x, y, p, q') 

sage: U = CoordinatePatch((x, y)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^2 with coordinates x, y 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y 

sage: x in F 

True 

sage: sin(y) in F 

True 

sage: p in F 

False 

sage: cos(q) in F 

False 

""" 

parent = None 

try: 

parent = element.parent() 

except AttributeError: 

pass 

if parent == self: 

return True 

if parent == SR: 

for coordinate in element.variables(): 

if coordinate not in self._patch.coordinates(): 

return False 

return True 

return False 

def _coerce_map_from_(self, S): 

""" 

Only the symbolic ring coerces into the algebra of differential forms. 

EXAMPLES:: 

sage: F = DifferentialForms(); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: F._coerce_map_from_(SR) 

True 

sage: F._coerce_map_from_(F) 

True 

sage: F._coerce_map_from_(CC) 

False 

sage: F._coerce_map_from_(RR) 

False 

""" 

return S is SR or S is self 

def _repr_(self): 

r""" 

String representation of this algebra of differential forms. 

EXAMPLES:: 

sage: x, y, z = var('x, y, z') 

sage: U = CoordinatePatch((x, y, z)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^3 with coordinates x, y, z 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: F._repr_() 

'Algebra of differential forms in the variables x, y, z' 

""" 

return "Algebra of differential forms in the variables " + \ 

', '.join(str(var) for var in self._patch.coordinates()) 

def _latex_(self): 

r""" 

Latex representation of this algebra of differential forms. 

EXAMPLES:: 

sage: x, y, z = var('x, y, z') 

sage: U = CoordinatePatch((x, y, z)); U 

doctest:...: DeprecationWarning: Use Manifold instead. 

See http://trac.sagemath.org/24444 for details. 

Open subset of R^3 with coordinates x, y, z 

sage: F = DifferentialForms(U); F 

doctest:...: DeprecationWarning: For the set of differential forms of 

degree p, use U.diff_form_module(p), where U is the base manifold 

(type U.diff_form_module? for details). 

See http://trac.sagemath.org/24444 for details. 

Algebra of differential forms in the variables x, y, z 

sage: latex(F) 

\Omega^\ast(\mathbb{\RR}^3) 

sage: latex(F) == F._latex_() 

True 

""" 

return "\\Omega^\\ast(\mathbb{\\RR}^%s)" % self._patch.dim()